Show that any positive odd integer is of the form 6q+1,or 6q+3, or 6q+5, where q is some integer .

Asked by gayatrivardani2002 | 6th Jun, 2016, 12:52: AM

Expert Answer:

B y space E u c l i d apostrophe s space d i v i s i o n space l e m m a space a n y space p o s i t i v e space i n t e g e r space a space i s space o f space t h e space f o r m comma
a equals b q plus r comma space 0 less or equal than r less than b
L e t space b equals 6
T h e n space w e space h a v e space a equals 6 q plus r comma space w h e r e space 0 less or equal than r less than 6
T h u s comma space r space c a n space t a k e space v a l u e s space 0 space t o space 5

W h e n space r equals 0 comma space
a equals 6 q
rightwards double arrow a equals 2 cross times open parentheses 3 q close parentheses
rightwards double arrow a equals 2 m comma space w h e r e space m equals 3 q
T h u s comma space a space i s space a n space e v e n space i n t e g e r

W h e n space r equals 1
a equals 6 q plus 1
rightwards double arrow a equals 2 cross times open parentheses 3 q close parentheses plus 1
rightwards double arrow a equals 2 m plus 1 comma space w h e r e space m equals 3 q
T h u s comma space a space i s space a n space o d d space i n t e g e r

W h e n space r equals 2
a equals 6 q plus 2
rightwards double arrow a equals 2 cross times open parentheses 3 q plus 1 close parentheses
T h u s comma space a space i s space a n space e v e n space i n t e g e r

W h e n space r equals 3
a equals 6 q plus 3
rightwards double arrow a equals 2 cross times open parentheses 3 q plus 1 close parentheses plus 1
T h u s comma space a space i s space a n space o d d space i n t e g e r

W h e n space r equals 4
a equals 6 q plus 4
rightwards double arrow a equals 2 cross times open parentheses 3 q plus 2 close parentheses
T h u s comma space a space i s space a n space e v e n space i n t e g e r

W h e n space r equals 5
a equals 6 q plus 5
rightwards double arrow a equals 2 cross times open parentheses 3 q plus 2 close parentheses plus 1
T h u s comma space a space i s space a n space o d d space i n t e g e r

A n y space i n t e g e r space c a n space b e space e i t h e r space o d d space o r space e v e n comma
T h u s comma space a n y space o d d d space p o s i t i v e space i n t e g e r space i s space o f space t h e space f o r m space space 6 m plus 1 comma space 6 m plus 3 space o r space 6 m plus 5 comma space m space i s space s o m e space i n t e g e r



space space space space space

Answered by  | 6th Jun, 2016, 06:11: PM

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